Assume X is a positive integer. If you divide 1/X, you will get a number that eventually becomes periodic: 1/9= 0.111..., 1/4= 0.25000..., and so on. Let's call numbers like 1/9 "pure" periodic, since the fractional part is formed just by the periodic part.

Prove that:

1. For all X, you will get a periodic part, and its length will be less than X.

2. If X is even, 1/X cannot be "pure". What happens if X is odd?

3. For some X, 1/X is "pure", the period length is even, and you can split the period in two halves that sum up to all nines. For example, 1/7=0.142857 142857... and 142+857=999. Which are these X values?

In a comment in Wacko Calculator (http://perplexus.info/show.php?pid=975&cid=6201) it is shown that you can express any pure repeating parts as geometric series, and thus, as (not necessarily reduced) fractions over all nines.

in Niners (http://perplexus.info/show.php?pid=231) it is shown that any number not divisible by 2 or 5 is a factor of one of these "all nines" numbers, and thus can be written as a pure 1/X, but since 2 and 5 are factors of powers of 10 (1 more than the niners), no niners will ever be divisible by a number with a factor of 2 or 5, so they can't be pure.

Posted by Gamer on 2006-10-16 13:43:46